J. Phys. Soc. Jpn. 88, 033001 (2019) [5 Pages]
LETTERS

Spontaneous Localization of Particles in Giant Vesicles Owing to Depletion Force

+ Affiliations
1Department of Mathematical and Physical Sciences, Faculty of Science, Japan Women’s University, Bunkyo, Tokyo 112-8681, Japan2Department of Creative Research, Exploratory Research Center on Life and Living Systems (ExCELLS), National Institutes of Natural Sciences, Okazaki, Aichi 444-8787, Japan3Department of Life and Coordination-Complex Molecular Science, Biomolecular Functions, Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki, Aichi 444-8585, Japan

When two kinds of particles with different sizes were encapsulated into identical giant vesicles, the small particles were localized near the inner membrane. The vesicles contained several hundreds of large particles and their volume fraction exceeded that of the small particles. Under the characteristic condition, where not only small but large particles are also considered as statistical ensembles contributing to the depletion force, the depletion force received by a small particle was estimated to be almost the same as that received by a large particle. In addition to this effect, small particles can be more accessible to the inner membrane.

©2019 The Author(s)
This article is published by the Physical Society of Japan under the terms of the Creative Commons Attribution 4.0 License. Any further distribution of this work must maintain attribution to the author(s) and the title of the article, journal citation, and DOI.

Entropic interaction is an important factor for characterizing the mesoscopic structure of soft matter;1,2) for example, colloidal hard particles aggregate in a solution of relatively small particles owing to the entropic interaction between the particles. This phase separation can be explained by the “Asakura–Oosawa (AO) theory”.3,4) In particular, a depletion region which excludes the centers of small particles, exists around large particles; however, when two large particles approach each other, this region is reduced owing to the overlap of their depletion regions. Consequently, this overlap of the depletion regions increases the free volume of the small particles (depletants). When the entropy of the small particles increases sufficiently owing to the increase in the free volume available to them, the aggregation of large particles occurs.

Furthermore, Dinsmore et al. demonstrated that this depletion force caused phase separation of polystyrene beads in a confined system.5) In their study, they considered large and small particles encapsulated into giant vesicles (GVs) under the condition that the small particles had a higher volume fraction (\(\varphi_{L} <\varphi_{S}\)) than that of the large particles. Thus, the large particles were localized spontaneously in the vicinity of the wall surface of the inner vesicular membrane. It has been confirmed that an increase in the free volume, where the centers of the small particle are accessible, also occurs because of the overlap of the depletion regions around the large particles with that around the membrane wall. However, for a closed system, wherein the volume fraction of large particles is considerably larger than that of small particles (\(\varphi_{L} >\varphi_{S}\)), the behavior of the internal particles has not yet been studied in sufficient detail.

We designed a GV system with large particles having a larger volume fraction than that of small particles (i.e., \(\varphi_{L} >\varphi_{S}\)).6) In particular, large polystyrene particles (diameter = 1.0 µm) were confined within the GVs at a higher volume fraction than small particles (diameter = 0.1 µm). As reported in our previous study,6) spontaneous localization of small particles is observed in the vicinity of the vesicular membrane; this phenomenon has been explained using the concept of a hypothetical boundary in the GVs, at which an equilibrium osmotic pressure is realized between an outer phase dominantly containing small particles and the corresponding inner phase. In the present work, we study this localization by considering the effective depletion force from the large particles as depletants. Thus, we present a novel understanding of GV systems, in which large particles push small particles toward the membrane wall.

The GVs were composed of 1,2-dioleoyl-\(sn\)-glycero-3-phosphocholine (DOPC), and were prepared using the water-in-oil emulsion centrifugation method to encapsulate the colloidal particles.68) The resulting GVs included large particles (diameter = 1.0 µm, composed of 5% green fluorescent polystyrene beads) and small particles (diameter = 0.10 µm, composed of 5% red fluorescent polystyrene beads). The inner aqueous phase of the obtained GVs was 0.15 M sucrose solution (0.02 M Tris–HCl buffer, pH 7.5), while the outer aqueous phase was 0.15 M glucose solution (0.02 M Tris–HCl buffer, pH 7.5). In our previous work,6) the volume fraction of the small spheres was 1.0 vol %, while multiple volume fractions were studied for the large spheres [Fig. 1(a)]. It is important to note that GVs with different volume fractions can be prepared simultaneously, because the inside particles are distributed to the divided vesicles asymmetrically during the process of centrifugation.


Figure 1. (Color) (a) Phase-contrast microscope image of the obtained GV dispersion. The white particles are polystyrene beads with a diameter of 1 µm. The GVs containing beads are indicated by yellow arrows. Using this preparation method, the GVs containing large particles at different volume fractions can be effectively obtained simultaneously. The red arrow indicates a GV with localized small particles. (b) Fluorescence microscope image showing the localization of small particles (red fluorescence beads) in the vicinity of the membrane wall. The analyzed GV is the same vesicle indicated by the red arrow in (a). (c) Plot profile of the average fluorescent intensity of a typical GV of (b). (d) Fluorescence microscope image showing large green fluorescence particles in the same GV. The corresponding video of the dynamic behavior owing to thermal motion in large particles present in the GV is provided in the Supplementary material.9) (e) Distribution of samples (red-filled circles) showing the localization of small particles for 4 h or longer after preparation. This figure is plotted with the volume fraction of large particles as the vertical axis and the diameter of GV as the horizontal axis. The error bars on the horizontal axis were evaluated according to Ref. 8.

As shown in Fig. 1(b), localization of the small particles was observed in the vicinity of the inner membrane of the GV. The fluorescence intensity profile of the representative fluorescence microscopy image [Fig. 1(b)] of the GV is shown in Fig. 1(c). According to the profile of the fluorescence intensity, small particles are localized with sub-micrometer width inside the GV. The Debye–Hückel screening length of our system was several nanometers, which was sufficiently smaller than the extent of this localization phenomenon. Fig. 1(d) shows the large green fluorescent particles present in the same GV as the one shown in Fig. 1(b); these large particles fluctuated randomly due to thermal motion (see Supplementary material),9) which suggests that these large particles can be considered as constituents of a thermal statistical ensemble. Therefore, we discuss the contribution of entropic interaction as a depletion force in this localization phenomenon.

In the scatter plot of the observed GVs [shown in Fig. 1(e)], each red-filled circle represents a GV with localized small particles. In Fig. 1(e), the horizontal axis corresponds to the diameter of a GV and the vertical axis corresponds to the volume fraction of the large particles.10) The GVs with diameters of 10 µm or more, in which localization of small particles was observed, were analyzed. From this plot, it can be seen that the localization of the small particles occurs frequently in the GVs in the case when the volume fraction of large particles is several times more than that of small particles. In particular, the localization occurs frequently in cases where a GV has a diameter of 13 µm or more with a 3 vol % or higher volume fraction for the large particles. It should be noted that the localization of small particles was frequently observed in a considerable number of GVs for 4 h or longer after preparation. Since the small particles remained localized for at least 2–3 weeks, this can be regarded as a configurational ensemble in a practical observation time. In the GVs where the localization of small particles was observed, the average value of the number of large particles \(N_{L}\) was 340. In addition, among the GVs wherein localization of small particles was observed, the GVs in which the number of small particles \(N_{S}\) was 30,000–40,000 were most frequently observed. Therefore, a typical specimen GV has the following properties: \(N_{L}\) is 300, \(N_{S}\) is 30,000, and the diameter is 15 µm. These values were used in the below-mentioned calculations.

In this study, we use the simple depletion interaction model to estimate the depletion force f that pushes the particles toward the inner surface of the GV membrane. Here, we define the particles extruded by depletants as “force-receptive particles”. In order to study the localization of the force-receptive particles, i.e., small particles in our experiment, we applied the AO theory to the cases where the volume fraction of the large particles (\(\varphi_{L}\)) is about one order of magnitude larger than that of the small particles (\(\varphi_{S}\)). Unlike the traditional AO theory experiments,5) it should be noted that a small particle is considered the force-receptive particle in the current study. In addition, the localization of the force-receptive particles is maintained by the self-depletion volume effect in the vicinity of the GV membrane wall because of two overlapping volumes,11) including that between the small particles and that between the small particles and the wall of the GVs. In order to investigate the mechanism through which only the small particles are localized in the vicinity of the vesicular membrane, we compare the corresponding depletion forces f of the small and large particles.

In this study, we consider the depletion force f, which is caused by a decrease in the depletion region because of the overlapping of the depletion regions. The magnitude of the depletion force f is the absolute value of the derivative of free energy G with respect to the distance D between the edge of the force-receptive particle and the wall.12) Here, G is the product of the osmotic pressure p and overlap volume \(V_{\text{over}}\): \begin{equation} \langle f\rangle_{T} = - \frac{\partial G}{\partial D} = - \frac{\partial V_{\text{over}}}{\partial D}\cdot p \end{equation} (1) We define \(V_{\text{over}}\) as the reduction in the volume of the depletion region because of the overlapping of the excluded volumes for the depletant around the force-receptive particles and around the wall (including the wall). Because the force-receptive particles always exclude the depletants from themselves, \(V_{\text{over}}\) does not include the volume of the force-receptive particles, and this volume should be subtracted from \(V_{\text{over}}\) depending on the distance between the force-receptive particle and the wall. We calculated the volumes of \(V_{\text{over}}\) for parts of the spherical shell by integral calculation with the center of the force-receptive particle as the origin. In our experiment, unlike in previous studies, the volume fraction of the large particles was higher than that of the small particles. Therefore, the role of the large particles as depletants cannot be ignored. When large particles act as force-receptive particles, both large as well as small particles can function as depletants. Similarly, when small particles act as force-receptive particles, both large as well as small particles can function as depletants. Therefore, combinations of force-receptive particles and depletants can be classified into the four cases listed in Table I. It should be noted that Cases (i) and (iv) are those wherein the depletant is pushed toward the wall by the particles of the same size (self-depletion), while Cases (ii) and (iii) are those wherein the particles are pushed toward the wall by particles of different sizes (i.e., depletion in the AO theory).

Data table
Table I. Combination of force-receptive particle and depletant in each case.

As shown in Fig. 2(a), we express the radii of a force-receptive particle and a depletant as \(R_{\textit{fr}}\) and \(R_{d}\), respectively; in addition, \(R_{\textit{fr}} + R_{d}\) is defined as L. The range of D is determined to be between \(2R_{d}\) and 0 in order to obtain a nonzero value for \(V_{\text{over}}\).


Figure 2. (Color online) (a) Schematic diagram of the depletion region in the vicinity of the membrane wall. The thickness of the exclusion region at the membrane and force-receptive particle surfaces is \(R_{d}\). (b) Results of numerical calculation of the depletion force f. Dependence of f on the distance D from the membrane wall to the particle surface. The lower solid line indicates the depletion force \(f_{S} = f_{(i)} + f_{(\textit{ii})}\) that the small particles (\(R_{S} = 0.05\) µm) receive, while the upper solid line indicates the depletion force \(f_{L} = f_{(\textit{iii})} + f_{(\textit{iv})}\) that the large particles (\(R_{L} = 0.5\) µm) receive. It is observed that \(f_{S}\) (D to \(R_{S}\)) and \(f_{L}\) (D to \(R_{L}\)) regions have almost similar values.

In Case (i), wherein both the force-receptive particle and the depletant are small particles, two regions, \(2R_{d} > D > R_{d}\) and \(R_{d} > D > 0\), should be considered. For the region \(2R_{d} > D > R_{d}\), we get \begin{equation} V_{\text{over}} = \pi \int_{R_{\textit{fr}} + D - R_{d}}^{L}(L^{2} - z^{2})\,dz. \end{equation} (2) Further, the region \(R_{d} > D > 0\) exists for systems wherein \(R_{d} > 2L/3\). Thus, we use the following expression: \begin{align} V_{\text{over}} &= \{\text{the right term of Eq. (2)}\} \notag\\ &\quad- \pi \int_{R_{\textit{fr}} + D - R_{d}}^{R_{\textit{fr}}}(R_{\textit{fr}}^{2} - z^{2})\,dz. \end{align} (3) It should be noted that \(V_{\text{over}}\) can be obtained as a function of D with various combinations of the radii \(R_{\textit{fr}}\) and \(R_{d}\). Because the force-receptive particle overlaps the depletion zone around the wall with a thickness of \(R_{d}\) [Fig. 2(a)], we subtract the overlapping volume of the force-receptive particles. By calculating the integrals in the above-mentioned equations, we obtain the following expression (here, \(R_{\textit{fr}}\) is written as \(L - R_{d}\)): \begin{equation} V_{\text{over}} = \begin{cases} \pi \left\{L^{2}[2R_{d} - D] - \dfrac{1}{3}[L^{3} + (2R_{d} - L - D)^{3}] \right\} &\text{for $2R_{d} > D > R_{d}$}\\ \pi \biggl\{L^{2}[2R_{d} - D] - \dfrac{1}{3}L^{3} - (L - R_{d})^{2}(R_{d} - D) + \dfrac{1}{3}(L - R_{d})^{3} \biggr\} &\text{for $R_{d} > D > 0$}\end{cases} \end{equation} (4) Then, we can obtain the partial derivative of \(V_{\text{over}}\) with respect to D as follows: \begin{equation} - \frac{\partial V_{\text{over}}}{\partial D} = \begin{cases} \pi \{L^{2} - (2R_{d} - L - D)^{2}\} &\text{for $2R_{d} > D > R_{d}$}\\ \pi \{L^{2} - (L - R_{d})^{2}\} &\text{for $R_{d} > D > 0$}\end{cases} \end{equation} (5) It should be noted that \(-\partial V_{\text{over}}/\partial D\) for \(R_{d} > D > 0\) is independent of D.

In Case (ii), wherein the force-receptive particle is a small particle while the depletant is a large particle, three regions, \(2R_{d} > D > R_{d}\), \(R_{d} > D > R_{d} - 2R_{\textit{fr}}\), and \(R_{d} - 2R_{\textit{fr}} > D > 0\), should be considered. The details of the calculation process are described in Ref. 13. In this case, we obtain the following equation. \begin{equation} V_{\text{over}} = \begin{cases} \pi \left\{L^{2}[2R_{d} - D] - \dfrac{1}{3}[L^{3} + (2R_{d} - L - D)^{3}] \right\} &\text{for $2R_{d} > D > R_{d}$}\\ \pi \left\{L^{2}[2R_{d} - D] - \dfrac{1}{3}L^{3} - (L - R_{d})^{2}(R_{d} - D) + \dfrac{1}{3}(L - R_{d})^{3} \right\} &\text{for $R_{d} > D > R_{d} - 2R_{\textit{fr}}$}\\ \pi \left\{L^{2}[2R_{d} - D] - \dfrac{1}{3}[L^{3} + (2R_{d} - L - D)^{3}] \right\} - \pi \cdot \dfrac{4}{3}(L - R_{d})^{3} &\text{for $R_{d} - 2R_{\textit{fr}} > D > 0$}\end{cases} \end{equation} (6)

Furthermore, we can obtain the partial derivative of \(V_{\text{over}}\) with respect to D as follows. \begin{equation} {-} \frac{\partial V_{\text{over}}}{\partial D} = \begin{cases} \pi \{L^{2} - (2R_{d} - L - D)^{2}\} &\text{for $2R_{d} > D > R_{d}$}\\ \pi \{L^{2} - (L - R_{d})^{2}\} &\text{for $R_{d} > D > R_{d} - 2R_{\textit{fr}}$}\\ \pi \{L^{2} - (2R_{d} - L - D)^{2}\} &\text{for $R_{d} - 2R_{\textit{fr}} > D > 0$}\end{cases} \end{equation} (7)

In contrast, we calculated \(V_{\text{over}}\) and \(-\partial V_{\text{over}}/\partial D\) of the two cases wherein the force-receptive particle is a large particle, i.e., Cases (iii) and (iv). Consequently, we found that \(V_{\text{over}}\) and \(-\partial V_{\text{over}}/\partial D\) in these two cases can be obtained using the same expressions as in Eqs. (4,) and (5), respectively.

As mentioned previously, we numerically estimate \(f_{S}\) and \(f_{L}\), which indicate the depletion forces for large particles (\(R_{L} = 0.5\,µ\text{m} = 10R_{s}\)) and small particles (\(R_{s} = 0.05\) µm), respectively, as described in Table I. Furthermore, as specified earlier in the previous part, the movement of the large particles can be considered to be a thermal motion with temperature T. Therefore, based on this assumption, we can treat the large particles as a part of a thermal statistical ensemble. Considering this, we concentrate our attention on the depletion force induced by \(-\partial V_{\text{over}}/\partial D\), which is averaged by such an ensemble.

The above-mentioned considerations based on thermal statistical mechanics can be used to obtain the following expression: \begin{equation} \langle f \rangle_{T} = - \frac{\partial V_{\text{over}}}{\partial D}\cdot p = - \frac{\partial V_{\text{over}}}{\partial D}\cdot\frac{N_{d}\cdot k_{\text{B}}T}{V_{0}}, \end{equation} (8) where p is the pressure of the statistical ensemble of the depletant. If such an ensemble can be treated as an ideal gas, the pressure can be expressed as \(p = (N_{d}\cdot k_{\text{B}}T)/V_{0}\), where \(V_{0}\) is the volume of the vesicle and \(N_{d}\) is the number of depletants. As mentioned in the experiments part, the number of large particles \(N_{L}\) is estimated to be about 300, as listed in Table I, whereas the number of small particles \(N_{S}\) is estimated to be about 30,000.

We calculated the depletion force \(f_{S}\), which is the sum of the depletion forces in Case (i) and (ii), and \(f_{L}\), which is the sum of the depletion forces in Cases (iii) and (iv); these forces are shown graphically in Fig. 2(b). It can be seen that the depletion forces for both the force-receptive large particles (\(f_{L}\)) as well as the small particles (\(f_{S}\)) attain their maximum values at \(D = R_{S}\). At this point, the ratio of the depletion forces is \(f_{L}\sim 7f_{S}\). However, as the particle size increases, it is more difficult for most of the force-receptive particles to enter the region within the radius \(D < R_{\textit{fr}}\). From Fig. 2(b), it can be deduced that, when the depletion forces in this region (\(D\sim R_{\textit{fr}}\)) are compared (\(f_{L}\) and \(f_{S}\)), the depletion force for one force-receptive large particle and that for one force-receptive small particle are considered to be similar [dotted-circle area in Fig. 2(b)].

In this study, two types of rigid polystyrene beads with different sizes were encapsulated in the same GVs. The large microspheres were encapsulated at a higher volume fraction than that of the small microspheres (\(\varphi_{L} >\varphi_{S}\)). Through our experiments, we observed that the small microspheres were localized in the vicinity of the inner surfaces of the vesicular membrane. This localization is suggested to be caused by the large volume fraction \(\varphi_{L}\) of the large particles compared with the volume fraction \(\varphi_{S}\) of the small particles in previously investigated systems. Although small particles also exert a force on the large particles, owing to the “AO” effect, the large volume fraction \(\varphi_{L}\) of the large particles suppresses such a behavior of the small particles. We concluded that this localization occurred because of the depletion force for the small (force-receptive) particles due to the overlapping of the depletion zones around the small particles and around the vesicle inner membrane. Consequently, the depletion force caused by the entropic interaction pushes the small particles to the surface of the vesicular inner membrane. In our experimental system, the large particles act as crowding agents called “depletants”; therefore, we treated these large particles as statistical ensembles.

In the present study, we pointed out the essential role of a depletant composed of large particles. We illustrated the scenario in which large particles push small particles in the direction of the wall. We should take into account the fact that the motion of the small particles is faster than that of the large particles in an environment where the volume fraction of the large particles is larger than that of the small particles. The crowding effect that occurs within the restricted space was derived from not only the overlapped volumes of various depletion regions, but also the effect of the diffusion coefficient on these particles, which is essentially determined by the sizes and masses of the particles.12) In fact, if we consider the thermal motion of a particle with mass m at temperature T, the average velocity v is estimated by \(v\sim \sqrt{3k_{\text{B}}T/m}\). In the present system, the velocity of the small particles was 32 times larger than that of the large ones, because the mass ratio of the large particles to small particles was approximately 1000. This fact suggests that the small particles can be promptly pushed to the wall in an environment with a high volume fraction of large particles.

Finally, we would like to highlight that the crowding effect in cells packed with biological macromolecules is caused by the effect of the diffusion coefficient on these biological macromolecules as well as the depletion volume effect; therefore, both these factors play an essential role in living organisms.12) Considering this, the experimental system used in our study is fundamental, and it is possible that the phenomena related to both of these effects are observed. Therefore, studying the underlying mechanism of the characteristic localization and elucidating its time scale might enable further understanding of this phenomenon.

Acknowledgment

This study was financially supported by the Okazaki ORION Project of the National Institutes of Natural Sciences, by JSPS KAKENHI Grant Numbers JP17K14374 and JP17H04876. Additional support was provided by a grant from the Kurita Water and Environment Foundation, and a grant from the Foundation, Oil & Fat Industry Kaikan.


References

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